// Order and Chaos in the Rotation and Revolution of Two Line Segments

Alex D. Saines

2011

Describing the motion of two line segments (slashes, //) orbiting about each other is the goal of the // Body Problem. To calculate the movement of the slashes in the // Body Problem we used the LaGrange method considering only the gravitational potential energy. Once an algebraic expression of the potential energy was obtained by integration of the Newtonian potential energy we created a Lagrangian for the system. Finally using the Euler-Lagrange equation we derived the exact equations of motion for all variables within the // Body Problem.

To check the potential energy we numerically integrated the potential at a specific set of parameters and then evaluated our derived potential energy using the same set of parameters and variables. We found that our potential energy is a near-exact version of the numerically integrated instance finding errors no larger than one part per hundred thousand using several tests. The tests include graphing the potential energy while rotating the slashes, moving the slashes physically apart while fixing their rotation, viewing their orbits and checking for the conservation of energy and momentum.

Further study included a preliminary search for families of orbits by numerically integrating the equations of motion. We attempted to find orbits which shared similar initial and final conditions based on the linear and angular velocities as well as the slashes separation. The orbits were further sorted by clarifying whether the specific initial conditions spawned an orbit where the slashes collided, diverged or formed stable orbits.